SAT Right triangles Question

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Question 1 · Hard · MathOctober 2024 Batch 5 · Q22Right triangles and trigonometry · 0 solved

$f (x) = a x^{2} + 2 x + c$ In the given quadratic function, a and c are constants. The graph of $y = f (x)$ in the xy-plane is a parabola that opens upward and has a vertex at the point $(h, k)$ , where h and k are constants. If $k < 0$ and $f (- 5) = f (1)$ , which of the following must be true? I. $c < 0$ I.a≥1

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Strategy

Use equal outputs to locate the axis of symmetry, then plug that x-value into the quadratic to relate the vertex height to the constants. Finally, check whether the given inequality forces each statement, or whether it only restricts a combination of terms.

Hints (open one at a time)

Hint 1

Equal outputs at two x-values mean those x-values are mirrored across the parabola’s axis of symmetry.

Hint 2

Find the midpoint of the two x-values to get the vertex’s x-coordinate, then evaluate the function there.

Hint 3

Use the condition on the vertex’s y-value to see what inequality it gives you about the constants, and decide whether that inequality forces each statement separately.

Step-by-step Explanation

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